Fairness-Aware Non-Orthogonal Multi-User Access With ... - IEEE Xplore

Received September 18, 2015, accepted November 19, 2015, date of publication December 7, 2015, date of current version January 6, 2016. Digital Object Identifier 10.1109/ACCESS.2015.2506261

Fairness-Aware Non-Orthogonal Multi-User Access With Discrete Hierarchical Modulation for 5G Cellular Relay Networks MEGUMI KANEKO, (Member, IEEE), HIROFUMI YAMAURA, YOUHEI KAJITA, KAZUNORI HAYASHI, (Member, IEEE), AND HIDEAKI SAKAI, (Life Fellow, IEEE) Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Corresponding author: M. Kaneko ([email protected]) This work was supported in part by the Japan Society for the Promotion of Science within the Grants-in-Aid for Scientific Research through the Ministry of Education, Science, Sports, and Culture of Japan under Grant 23760334, Grant 26820143, Grant 15H2252, and Grant 15K06064, and in part by the Telecommunications Advancement Foundation.

ABSTRACT Non-orthogonal multiple access based on superposition coding (SC) for 5G cellular systems without relays is gaining increasing interest from both academia and industries. Since relay stations will be an integral part of future cellular networks, we propose evolved non-orthogonal multi-access schemes for both direct and relayed users. Our schedulers are built upon SC-relaying schemes with practical discrete hierarchical modulations (HMs), where the messages of two selected users are superposed into different HM layers, each layer being allocated an optimized amount of power and bearing a message flow to be decoded through either a direct or relayed link. As opposed to conventional schedulers that allocate orthogonal resources to each user in wireless relaying systems, our non-orthogonal schedulers allow a pair of selected users to simultaneously share their allocated resource unit. Moreover, unlike the SC-relaying schemes in the literature based on Gaussian codebooks, the proposed schemes are designed and analyzed under the practical constraints of discrete HMs. In spite of the complexity of the power optimization under discrete HMs, we provide a simple and near-optimal power allocation method for sum-rate maximization and proportional fairness. The simulation results show that the conventional orthogonal schedulers are outperformed by the proposed schedulers in terms of sum rate and fairness, even under the practical assumption of HMs. INDEX TERMS Cellular relay system, radio resource allocation, superposition coding, hierarchical modulation, non-orthogonal multiple access. I. INTRODUCTION

Many recent research works have demonstrated the efficiency of relaying techniques for wireless communications such as Multi-Hop (MH) and Cooperative Diversity (CD) transmissions [1], [2], providing various advantages such as rate improvement and coverage extension with only low deployment costs [3], [4]. For the single-user relay channel, [5] has proposed a relaying scheme based on Superposition Coding (SC) where two superposed messages in the modulation domain are sent through the direct and relayed paths and recovered by Successive Interference Cancelation (SIC) at the destination, improving the achievable rates of MH and CD. A crucial aspect of cellular relay systems concerns the design of efficient scheduling algorithms as surveyed in [6] which can be a very complex problem given the large

2922

number of possible sender to destination paths. Most existing algorithms for relayed systems as in [6]–[8] allocate an orthogonal resource, which can be in time, frequency, code or space, to each user within a cell or sector, i.e., each Base Station (BS) or Relay Station (RS) serves only a unique user per resource unit. However, the suboptimality of orthogonal allocation is an established fact, as higher spectral efficiency may be achieved with non-orthogonal allocation where multiple users are simultaneously served by one access point over each resource unit [9], [10]. This is the principle of SC with SIC, the capacity-achieving scheme in the Gaussian Broadcast Channel (GBC) [10], where the messages to multiple users are superposed with an appropriate power ratio [11]. The principle of Non-Orthogonal Multiple Access (NOMA) based on SC is gaining more and more interests not only in

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VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

the research community but also in the industry, such as NTT Docomo which is actively developing the NOMA concept based on SC for the Future Radio Access in the 2020s, for 5G cellular systems without relays [12], [13]. Given the generalized deployment of relays for next generation systems, it can be naturally foreseen that NOMA based on SC for relay-aided systems will be the next major milestone. Thus, non-orthogonal SC schedulers for cellular relay systems were designed in [14]–[17], built upon a generic scheme designed for the two-user Relay Broadcast Channel (RBC) of Fig. 1 in which two users, MS1 and MS2 , are simultaneously served by a BS with the help of a RS. In particular, the scheme proposed in [14] superposes the two users’ messages into three SC layers, exploiting one high quality direct link (e.g., BS-MS1 in Fig. 1) and the two relayed links (BS-RS-MS1 and BS-RS-MS2 ). In that scheme, power ratios are optimized over the three SC layers to maximize the sum rate [15] or Proportional Fairness (PF) criterion [16]. It was shown that these schedulers outperform conventional non-orthogonal schedulers in terms of sum-rate, outage probability and user fairness. However, all these non-orthogonal schedulers and their underlying SC relaying schemes have been designed under the assumption of Gaussian codebooks. As practical wireless relay systems make use of discrete modulations, the schemes and power allocation optimization in [14]–[17] are not applicable. Thus, feasible SC schemes are strongly required to be designed taking into account these practical constraints.

FIGURE 1. Two-user relay broadcast channel.

The aim of this work is to design SC relaying schemes and non-orthogonal schedulers making use of discrete Hierarchical Modulations (HM)s [18]–[20], which is necessary for their implementation into next generation relayaided cellular systems. HM consists of non-uniformly spaced constellation points, providing different levels of error protection to messages superposed in the same symbol. Given the discrete nature of HMs, the whole problem setup is totally different compared to the previous schemes designed under the Gaussian codebook assumptions. This is why new scheduling design and power allocation analysis for this evolved problem setup are required. VOLUME 3, 2015

A. RELATED WORKS ON HM-BASED RELAYING AND/OR SCHEDULING

Several works have considered the design of HM schemes for various relaying and scheduling scenarios. In [21], a coded cooperation scheme based on HM has been proposed where two users cooperate through their direct link. The problem of multi-user scheduling with HM schemes has been considered in [11]–[13] and [22], but in a downlink system without any relays. HM in the single-user relay channel was proposed in [23], with two types of HM levels, whereas [24], [25] designed HM schemes with optimized power allocation among the HM layers. For the same channel, [26] proposed an HM-based cooperative scheme for Symbol Error Rate (SER) minimization, while optimal Signal-to-Noise Ratio (SNR) threshold switching between two types of HMs was designed in [27]. Then, [28] applied the two-layer HM scheme for the two-user RBC, which is shown to outperform the scheme without relaying in terms of SER. However, no power optimization is performed. Finally, other HM-based schemes can be found for interference mitigation purposes as in [29], or for a single source/destination channel aided by multiple relays [30]. Note also that the NOMA concept by NTT Docomo [12] which is currently attracting huge research interests has only dealt with SC in cellular systems without relaying so far.

B. OUR NEW CONTRIBUTIONS

In this work, we first propose a generic two-user SC scheme based on discrete HMs, the Three-Layer Two-User HM (3L2UHM) scheme, that simultaneously serves two users whose messages are superposed into three discrete HM layers with specific power allocation ratios. Two different objective functions are considered in the analysis of power allocation optimization: Sum-Rate (SR) and General Proportional Fairness (GPF) maximization. Next, each of these schemes are integrated into the proposed nonorthogonal schedulers for the Downlink (DL) of a multi-user cellular relay system, namely the Two-User Sum-Rate Maximizing (TU-SRM) scheduler which selects the sum-rate maximizing users per scheduling frame, and the Two-User GPF Maximizing (TU-GPFM) scheduler that serves the user pair whose GPF metric is maximal. The proposed non-orthogonal multi-user access schemes built upon these generic three-layer HM schemes provide an additional nonorthogonal option for the scheduler compared to previous orthogonal access schemes. A low-complexity version of the TU-GPFM scheduler is also designed. Thus, the main advantages of our schemes and new contributions can be given as follows: - As opposed to the previous works limited to two HM levels for relay channels, our scheme comprises three HM levels with optimized power ratios, allowing to further performance enhancement by taking full advantage of the three best link flows. Although a maximum of four link flows could be exploited in a two-user RBC, a four-level HM would lead 2923

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

to a very complex scheme with marginal performance gain.1 Therefore, the proposed relaying scheme based on three-level HMs allows the best compromise between performance and feasibility. - Power optimization among the three HM levels is performed for two different metrics: sum-rate and PF. The simulation results show that, for the two-user RBC, the proposed generic schemes outperform the rate and GPF performance compared to conventional relaying schemes. - Unlike the previous works on HM-based relaying which mainly considered a one or two user system, our systemlevel simulations show the effectiveness of the proposed non-orthogonal schedulers both in terms of sum-rate and GPF, compared to the conventional orthogonal schedulers which allocate each resource unit to a unique user, while the proposed low-complexity algorithm achieves an excellent trade-off between the involved performance metrics. - Most importantly, we have confirmed that even under the more stringent constraints of discrete HMs, non-orthogonal schedulers outperform traditional orthogonal schedulers in terms of various system level metrics, similarly to the main conclusions of previous works [15], [17] that assumed Gaussian codebooks. Thus, our results open up new perspectives towards the integration of non-orthogonal multiple access into next-generation cellular relay systems. The sequel of the paper is organized as follows. The system model is introduced in Section II, then the reference schemes in Section III. The proposed 3L2UHM scheme and power allocation analysis are described in Section IV, then the proposed schedulers in Section V. Numerical results are presented in Section VI. Finally, conclusions are given in Section VII.

assuming orthogonalized resources in frequency or time among sectors. BS, RS and MS are each equipped with a single antenna. The half-duplex RS works under the Decodeand-Forward (DF) protocol and SIC detector. The direct and relayed links refer to the BS-MS and BS-RS-MS links, respectively. In each scheduling frame composed of the two steps below, one or two users are served through the BS, the RS, or both depending on the relaying scheme. The two user system scheduled in each frame by the proposed non-orthogonal schedulers constitute the TU-RBC in Fig. 1. In Step 1, the BS transmits a vector of N symbols x = [x(1), . . . , x(N )]T . The received signals at the RS and MSi , i = 1, 2 are given by yR = hR x + zR , yDi = hDi x + zDi ,

(1)

respectively. In Step 2, the RS transmits a vector of NR symbols xR = [xR (1), . . . , xR (NR )]. The received signal at MSi , i = 1, 2 is given by yRi = hRi xR + zRi .

(2)

In (1)-(2), hI , I ∈ {R, Di, Ri}, i = 1, 2, denote the complex channel coefficients of links BS-RS, BS-MSi , RS-MSi , while zI are vectors of a circular-symmetric complex Additive White Gaussian Noise (AWGN) whose elements have mean zero and variance σ 2 . The instantaneous link SNRs are defined as γI =

|hI |2 , σ2

I ∈ {R, Di, Ri},

i = 1, 2.

(3)

TABLE 1. Discrete modulation set.

FIGURE 2. Cellular multi-user relay system.

II. SYSTEM MODEL

We consider the DL multi-user cellular relay system in Fig. 2, where each cell contains three sectors with one RS each. Scheduling is performed independently for each sector, 1 Note that it was also proved in [17] that using the three best links, and

hence three-layer SC scheme was sum-rate optimal. 2924

All channel coefficients hI , I ∈ {R, Di, Ri}, are assumed to be constant during each transmission time frame and to be known by the BS and the RS before transmission. We assume transmitted symbols to have mean zero E[x(n)] = E[xR (nR )] = 0, and an average power of E[|x(n)|2 ] = E[|xR (nR )|2 ] = 1. The bandwidth of transmitted signals is assumed to be normalized to 1. We consider the discrete modulations: BPSK, QPSK, 8-PAM, 16-QAM and 64-QAM as in Table 1 where m = 1, . . . , 5 is the modulation level with rate r(m).2 We also define the two-layer Hierarchical QAM (HQAM): [BPSK]2 -HQAM in Fig. 3, [QPSK]2 -HQAM in Fig. 4 and similarly, [8-PAM]2 -HQAM as explained in Table 2 where m ¯ = 1, 2, 3 is the HM level. 2 Note that we have considered the discrete modulations and HMs in Tables 1 and 2 without loss of generality, i.e., the proposed schemes may be applied to different sets of modulations. VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

FIGURE 3. [BPSK]2 -HQAM constellation.

 1p 1   αx (2 − 3αx ) < αy < 1 − αx ,  (1 − αx ) +   2 2  1 2   for < αx < , 2 3 (4) 1   (1 − α ) < α < 1 − α ,  x y x  2    2   for ≤ αx < 1. 3 Thus in Fig. 5, the hierarchical constellation points Si (i = 1, . . . , 16) generated from the QPSK symbol Sf in the first layer stay within the first quadrant. For αx = 16/21 and αy = 4/21, [QPSK]3 -HQAM has a square constellation, i.e., the same one as for 64-QAM. III. REFERENCE TRANSMISSION SCHEMES AND SCHEDULERS

We describe the reference SU and TU transmission schemes used in reference schedulers. A. REFERENCE SU AND TU TRANSMISSION SCHEMES 1) DIRECT TRANSMISSION (DT) SCHEME

FIGURE 4. [QPSK]2 -HQAM constellation. TABLE 2. Discrete two-layer HM set.

During both steps 1 and 2 described in Section II, BS transmits to an MSi directly without any help from the RS. The rate is defined as the number of correct bits received by MS per unit symbol time, where the whole message is discarded if at least one bit in the message is not decoded correctly. The rate of DT scheme is obtained as RDT = r(m) (1 − P1 (m, γDi ))N , where P1 (m, γ ) is the Symbol Error Rate (SER) for modulation level
m and SNR γ . Defining erfc(x) = R∞ 2 dt, the SER is given by [31] √2 exp −t π x  1 √
  erfc γ , m = 1,   2   r 2    1 γ   1 − 1 − erfc , m = 2,   2   r 2   7 γ , m = 3, (5) P1 (m, γ ) = 8 erfc 21   r    2   3 γ    1 − 1 − erfc , m = 4,   4 10   r    2   7 γ   1 − 1 − erfc , m = 5. 8 42 2) MULTI-HOP (MH) SCHEME

FIGURE 5. [QPSK]3 -HQAM constellation.

Moreover, we introduce [QPSK]3 -HQAM which is a three-layer HQAM created by superimposing three QPSK symbols, as shown in Fig. 5 where three QPSK symbols have power αx , αy and αz = 1−αx −αy in HM layers 1, 2 and 3, respectively. We assume the following conditions for αx and αy so that each constellation point stays within the same quadrant as its corresponding QPSK symbol in the first layer, VOLUME 3, 2015

Only the relayed signal is considered at MSi , not the direct one. After RS decodes the received signal from the BS in Step 1, it forwards in Step 2 the remodulated signal given SNR γRi . BS selects modulation level m1 in Step 1, while RS uses modulation level m2 in Step 2. In Step 2, the RS forwards Nr(m1 ) bits to the MS with the rate r(m2 ). Thus, 1) the duration of Step 2 is Nr(m r(m2 ) . The rate of MH scheme is RMH =

Nr(m1 ) N+

Nr(m1 ) r(m2 )

(1 − P1 (m1 , γR ))N

× (1 − P1 (m2 , γRi ))

Nr(m1 ) r(m2 )

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3) COOPERATIVE DIVERSITY (CD) SCHEME

In the CD scheme of [32], different modulation rates can be used at the BS and the RS. In Step 1, the signal transmitted by the BS with modulation level m1 is received by both RS and MSi . At RS, the received signal is decoded and retransmitted to MSi in Step 2 using the modulation adapted to SNR γRi . If RS uses the same modulation as the BS, MSi performs MRC of the signal received from BS in Step 1 with the one received from RS in Step 2 and decodes the combined signal. Otherwise, RS uses modulation level m2 6 = m1 and MSi decodes his message by combining the received signals from the BS and the RS by using Log-Likelihood Ratio (LLR) combining for each bit. When m1 = m2 , the duration of Step 1 is equal to that of Step 2. Thus, the rate is RCD =

r(m1 ) (1 − P1 (m1 , γR ))N (1 − P1 (m1 , γDi + γRi ))N . 2

1) When m1 6 = m2 , the duration of Step 2 is Nr(m r(m2 ) . In this case, the SNR of the combined signal is no longer given by the sum of the SNRs of the direct and the relayed signals. To solve this problem, we use the approximated   Bit Error Rate (BER) bγ expression BER(γ ) = aexp − k(m) in [33] and [34], where a and b are the fitting parameters for the BER of BPSK and k(m) is a specific parameter for modulation level m which is equal to one in the case of BPSK. We obtain k(m) = 1, 2, 21, 10, 42 for m = 1, 2, 3, 4, 5, respectively, where a = 0.268 and b = 1.0358 were determined as least square solutions in logarithmic scale, respectively. It was  found that  γDi γRi the expression BER = aexp −b k(m + gave a k(m2 ) 1)

very good approximation of the BER of the combined signal at the MS, giving the rate RCD =

Nr(m1 )

(1 − P1 (m1 , γR ))N

1) N + Nr(m r(m2 )    Nr(m1 ) γRi γDi × 1 − aexp −b + . k(m1 ) k(m2 )

4) SINGLE-USER SUPERPOSITION CODING (SUSC) SCHEME

In the√scheme of √ [24] and [25], in Step 1, BS generates x = 1 − αxb + αxs using [BPSK]2 -HQAM, [QPSK]2 HQAM or [8-PAM]2 -HQAM, where α ∈ (0, 1) is the power allocation parameter between the basic and the superposed symbols xb and xs intended √ for MS. RS decodes xb from the received signal, treating αxs as noise. Subtracting √ 1 − αxb from the received signal, RS decodes xs , while MS keeps the received signal from the BS in memory. In Step 2, RS transmits xR , the remodulated signal of xs correctly decoded in Step 1, using the modulation in Table 1 adapted to SNR γR1 . MS decodes xR (xs ) from the signal received from RS and cancels its contribution from the signal kept in Step 1, then decodes xb . BS selects HM level m ¯ 1 in Step 1, while RS uses level m2 in Step 2. The rate of SUSC scheme is 2926

obtained as r(m ¯ 1 )r(m2 ) {1 − P2 (m ¯ 1 , γR , 1 − α, α)}N r(m ¯ 1 ) + r(m2 )

RSUSC =

× {1 − P1 (m ¯ 1 , γR α)}N {1 − P1 (m2 , γR1 )} h i × 1 + {1 − P1 (m ¯ 1 , γD1 (1 − α))}N .

Nr(m ¯ 1) r(m2 )

In (6), P2 (m, ¯ γ , α1 , α2 ) is the SER of the symbol in the first layer of two-layer HM with SNR γ , and α1 , α2 are the power ratios allocated to the symbols in the first and second layers, respectively. The SER P2 (m, ¯ γ , α1 , α2 ) is given by P2 (m, ¯ γ , α1 , α2 ) 1
√  erfc γ α1 ,    2  r r    γ α1 γ α2 1    1 − 1 − erfc −  4 2 2 r r 2 = 1 γ α γ α  1 2   − erfc + ,   4  2 2  r     γ α1 7   erfc , 8 21

for m ¯ = 1,

for m ¯ = 2, for m ¯ = 3. (6)

5) TWO-LAYER TWO-USER SUPERPOSITION CODING (2L2USC) SCHEME

An initial two-user SC based scheme was proposed in [35] for the TU-RBC as in Fig. 1 assuming Gaussian codebooks. Here, we introduce the reference 2L2USC scheme with discrete HM based on the scheme in [35]. It simply follows SUSC by setting xb = x1 destined to MS1 and xs = x2 destined to MS2 . Thus, the sum rate of 2L2USC scheme is R2L2USC =

r(m ¯ 1 )r(m2 ) {1 − P2 (m ¯ 1 , γR , 1 − α, α)}N r(m ¯ 1 ) + r(m2 ) × {1 − P1 (m ¯ 1 , γR α)}N {1 − P1 (m2 , γR2 )} h i × 1 + {1 − P1 (m ¯ 1 , γD1 (1 − α))}N .

Nr(m ¯ 1) r(m2 )

B. REFERENCE SINGLE-USER SCHEDULERS

In the DL multi-user cellular relay system, conventional scheduling algorithms allocate orthogonal channels (typically, different time slots/frequency channels) to different users, where the user with the corresponding reference single user transmission scheme of Section III-A that achieves the best target metric is served in each frame. We consider sum-rate maximization and proportional fairness in the schedulers described below. 1) Single User Sum-Rate Maximizing (SU-SRM) Scheduler: For each user k, (k = 1, . . . , K ), the scheduler selects the SU transmission scheme among DT, MH, CD and SUSC that achieves the highest rate. Let Rk be the rate of user k in the current frame, with the selected best SU scheme. Then, the user k ∗ with the highest rate is selected to be scheduled in the current frame, k ∗ = arg maxk Rk . VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

2) SU Normalized Proportional Fairness Maximizing (SU-NPFM) Scheduler: as shown in [36], the PF scheduler is defined as the scheduler S that maximizes the GPF metric K X (S ) 0S = log Rk , (7) k=1 (S )

where Rk denotes user k’s achievable rate with scheduler S. In the SU case, one conventional PF scheduler [37] is the one that selects the user maximizing the Normalized PF (NPF) metric Rk , (8) ρk = R¯ k where R¯ k is user k’s long-term average rate given by its average SNR. Therefore, the Reference SU-NPFM scheduler works as follows: after selecting the best SU transmission scheme for each user as in SU-SRM scheduler, it chooses the user k ∗ with the highest NPF metric in each frame, i.e., k ∗ = arg maxk ρk .

The proposed scheme may be applied whenever the link SNRs satisfy γR ≥ γRi ≥ γDi , i = 1, 2 or, γR ≥ γRi ≥ γDi and γR ≥ γDj ≥ γRj for i 6 = j. Therefore in the analysis, the following link SNR ordering will be assumed without loss of generality for the proposed scheme γD2 ≤ γD1 ≤ γR1 ≤ γR2 ≤ γR ,

(11)

as the analysis applies in all cases by adapting the SIC decoding orders at each receiver as follows. That is, as explained in [17] and the SC scheme in [9] for broadcast channel, we allocate more power to the messages with weaker link qualities for improving user fairness. Since at the final decoding stage, xb1 , xs1 , x2 are message flows over the links with SNRs satisfying γD1 ≤ γR1 ≤ γR2 , this implies the power ratio constraint αb1 ≥ αs1 ≥ α2 for ensuring higher power to the messages on weaker links. Thus, the optimal SIC decoding order at RS is xb1 → xs1 → x2 , since the message with highest power should be successively decoded and canceled, following the SIC principle. Thus, RS decodes xb1 from yR , treating the contributions of xs1 and x2 as noise. Subtracting √ √ αb1 xb1 from yR , we obtain y0R = yR − hR αb1 xb1 = √ √ hR ( αs1 xs1 + α2 x2 ) + zR , from which the RS decodes xs1 , √ √ treating hR α2 x2 as noise, giving y00R = y0R − hR αs1 xs1 = √ hR α2 x2 + zR . The RS finally decodes x2 from y00R . On the other hand, MS1 receives √ √ √ yD1 = hD1 ( αb1 xb1 + αs1 xs1 + α2 x2 ) + zD1 , (12) and keeps it in memory. As the link SNR γD2 is the worst, MS2 ignores its received signal. Step 2: The RS transmits the two-layer hierarchically modulated signal p p (13) xR = 1 − βxR1 + βxR2 ,

FIGURE 6. Steps of the proposed scheme.

IV. PROPOSED RELAYING SCHEMES WITH HM A. DESCRIPTION OF THE STEPS

We describe the proposed Three-Layer Two-User HM (3L2UHM) for the TU-RBC in Fig. 6. Step 1: Using [QPSK]3 -HQAM, BS generates the three-layer hierarchically modulated signal √ √ √ x = αb1 xb1 + αs1 xs1 + α2 x2 , (9) where αb1 , αs1 , α2 ∈ [0, 1] denote the power allocation parameters of HM layers 1, 2 and 3, respectively, summing up to one, αb1 + αs1 + α2 = 1. xb1 and xs1 are destined to MS1 and x2 to MS2 . The RS receives √ √ √ yR = hR ( αb1 xb1 + αs1 xs1 + α2 x2 ) + zR . (10) VOLUME 3, 2015

created with [BPSK]2 -HQAM, [QPSK]2 -HQAM or [8-PAM]2 -HQAM. xR1 and xR2 are the remodulated signals of xs1 and x2 , respectively. β ∈ (0, 1) is the power allocation parameter. The received signal at MS2 is p p yR2 = hR2 ( 1 − βxR1 + βxR2 ) + zR2 . (14) Since γR1 ≤ γR2 , MS2 first decodes xR1 then xR2 through SIC. From yR2 , MS2 decodes xR1 and hence xs1 . Canceling the √ component of xR1 √ from yR2 , MS2 obtains y0R2 = yR2 − hR2 1 − βxR1 = hR2 βxR2 + zR2 , from which it decodes xR2 (x2 ). Meanwhile, MS1 receives p p yR1 = hR1 ( 1 − βxR1 + βxR2 ) + zR1 . (15) √ MS1 decodes xR1 from yR1 , treating hR1 βxR2 as noise. Thus, MS1 obtains xs1 and cancels xs1 from yD1 kept in √ memory in Step 1, getting y0D1 = yD1 − hD1 αs1 xs1 = √ hD1 αb1 xb1 + zD1 , from which MS1 finally decodes xb1 . B. EXPRESSIONS FOR SUM RATE AND GPF METRIC

In this section, we derive the sum rate of the proposed 3L2UHM scheme defined as the sum of the expected 2927

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

number of correctly received bits at MS1 and MS2 per unit symbol time. It is assumed that if there is at least one bit error in a decoded message, the whole message is discarded. (R) Let x¯ i (i ∈ {b1, s1, 2}) be the message xi decoded at the RS. (R) We define Qb1 as the probability of correct decoding of (R) message xb1 by RS, and Qs1 as the probability of correct decoding of message xs1 given the correct decoding of xb1 by RS, where we have denoted the conditional probability (R) of event A given B as Pr(A | B). Similarly, Q2 is defined as the probability of correct decoding of x2 given the correct decoding of xb1 and xs1 by RS, namely (R)

(R)

(R)

(R)

(R) Q2

(R) Pr(¯x2

On the other hand, the GPF metric defined in (7) is given by 03L2UHM = log RMS1 + log RMS2 = log RMS1 RMS2 ,

so maximizing 03L2UHM is equivalent to maximizing the user rates’ product RMS1 RMS2 ,  RMS1 RMS2 =

(MS1 )

×QR1

(MSj )

= x2 |

(R) x¯ s1

=

(i ∈ {b1, s1, 2}, j ∈ {1, 2}) be the message xi

(MS ) QR1 1

(MS ) and Qb1 1 of correct decoding of the at MS1 , MS2 similarly as above, giving

(MS2 )

= Pr(¯xR1

(MS2 )

= Pr(¯xR2

(MS1 )

= Pr(¯xR1

(MS1 )

= Pr(¯xb1

QR2

QR1 Qb1

(MS2 )

= xR1 | x¯ (R) = x),

(MS2 )

= xR2 | x¯ R1

(MS1 )

= xR1 | x¯ (R) = x),

(MS1 )

= xb1 | x¯ R1

(MS2 )

(MS1 )

(MS2 )

, QR2

,

involved mes-

= xR1 , x¯ (R) = x), = xR1 , x¯ (R) = x),

where x¯ (R) = x represents the correct decoding of all messages xb1 , xs1 and x2 at RS in Step 1. Since BS uses [QPSK]3 -HQAM in Step 1, each message xi (i ∈ {b1, s1, 2}) is composed of 2N bits where N is the number of symbols per message block. Message xs1 is correctly decoded at MS1 if all messages xb1 , xs1 and x2 were correctly decoded at the RS, and if xs1 was correctly decoded at MS1 . Thus, the expected number of bits of xs1 that MS1 cor(R) (R) (R) (MS ) rectly decodes is 2NQb1 Qs1 Q2 QR1 1 . Similarly, the expected number of bits of xb1 that MS1 correctly (R) (R) (R) (MS ) (MS ) decodes is 2NQb1 Qs1 Q2 QR1 1 Qb1 1 , and the expected number of bits of x2 that MS2 correctly decodes is (R) (R) (R) (MS ) (MS ) 2NQb1 Qs1 Q2 QR1 2 QR2 2 . Since BS transmits N symbols in Step 1, and RS NR symbols in Step 2, the total duration is N + NR symbol times. Thus, the achievable rates of MS1 and MS2 can be written as RMS1 RMS2

2N (R) (R) (R) (MS ) (MS ) = Q Q Q Q 1 (1 + Qb1 1 ), N + NR b1 s1 2 R1 2N (R) (R) (R) (MS ) (MS ) = Q Q Q Q 2 QR2 2 , N + NR b1 s1 2 R1

(16) (17)

(MS1 )

2928

(20)

(MS1 )

(1+Qb1

(MS2 )

)+QR1

(MS2 )

QR2

o

. (18)

(21)

P3 (γ , αb1 , αs1 ) r r  r  γ αb1 γ αs1 γ α2 1 = 1 − 1 − erfc − − 8 2 2 2 r r r  1 γ αb1 γ αs1 γ α2 − + − erfc 8 2 2 2 r r r  1 γ αb1 γ αs1 γ α2 − erfc + − 8 2 2 2 r r r 2 1 γ αb1 γ αs1 γ α2 − erfc + + . 8 2 2 2 Next, after canceling xb1 from the received signal yR in Step 1, the resulting signal y0R is regarded as a [QPSK]2 -HQAM (R) symbol with AWGN. Thus, Qs1 is obtained as (R)

Qs1 = {1 − P2 (2, γR , αs1 , α2 )}N ,

(22)

where SER P2 (m, ¯ γ , α1 , α2 ) was given in (6). Then, y00R obtained by canceling xs1 from y0R is a QPSK symbol with AWGN, giving (R)

Q2 = {1 − P1 (2, γR α2 )}N ,

(23)

where SER P1 (m, γ ) was given in (5). (MS ) (MS ) (MS ) In Step 2, if RS uses HM level m ¯ 2 , QR1 2 , QR2 2 , QR1 1 (MS1 )

and Qb1

can be written as

(MS2 )

= {1 − P2 (m ¯ 2 , γR2 , β, 1 − β)}NR ,

(MS2 )

= {1 − P1 (m ¯ 2 , γR2 β)}NR ,

(MS1 )

= {1 − P2 (m ¯ 2 , γR1 , β, 1 − β)}NR ,

(MS1 )

= {1 − P2 (2, γD1 , αb1 , α2 )}N .

QR1 QR2 Qb1

2N (R) (R) (R) Q Q Q N n+ NR b1 s1 2

× QR1

.

where P3 is the SER of the symbol in the first layer of [QPSK]3 -HQAM, determined as

QR1

and hence the overall sum-rate as R3L2UHM =

(MS2 )

QR2

Qb1 = {1 − P3 (γR , αb1 , αs1 )}N ,

= xb1 ).

(MS2 )

QR1

(MS2 )

)QR1

(R)

(R) xs1 , x¯ b1

decoded at MSj . We also define probabilities QR1 sages

(MS1 )

(1 + Qb1

2

(R)

Qs1 = Pr(¯xs1 = xs1 | x¯ b1 = xb1 ),

Let x¯ i

2N (R) (R) (R) Q Q Q N + NR b1 s1 2

Next, we derive the expressions of each term in R3L2UHM and (R) 03L2UHM . First, Qb1 is given by

Qb1 = Pr(¯xb1 = xb1 ),

=

(19)

(24)

Finally, the analytical expressions of R3L2UHM in (18) and 03L2UHM in (19) are obtained by substituting these probabilities by their expressions in (21), (22), (23) and (24). VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

C. POWER ALLOCATION OPTIMIZATION FOR SUM-RATE MAXIMIZATION

We perform the power allocation optimization for maximizing the sum-rate (18). To achieve the global solution, all parameters αb1 , αs1 and β should be jointly optimized. However, this problem is very difficult due to the complexity and non-convexity of the sum rate expression, i.e., optimality can only be guaranteed by exhaustive search over the three dimensions, which is very time-consuming. Instead, we take a suboptimal approach where we separate the optimization in two steps, first on β and then on (αb1 , αs1 ), as (αb1 , αs1 ) determine the power allocation in the first step, and β that of the second step. This approach will be compared to the optimal solution obtained by exhaustive search. 1) OPTIMIZING αb1 AND αs1 FOR GIVEN β

To determine the values of αb1 and αs1 that maximize R3L2UHM for given β, we differentiate the sum-rate expression in (18) with respect to αb1 and αs1 . Denoting I ∈ {b1, s1}, we get ∂R3L2UHM ∂α " I (R) # (R) (R) ∂Qb1 (R) (R) (R) ∂Qs1 (R) (R) (R) ∂Q2 = Q Q + Qb1 Q + Qb1 Qs1 ∂αI s1 2 ∂αI 2 ∂αI (MS1 ) n   o (MS ) (R) (R) (R) ∂Qb1 × C1 1 + Qb1 1 +C2 + C1 Qb1 Qs1 Q2 , ∂αI (25) (MS )

1 and C2 = N 2N where the constants C1 = N 2N +NR QR1 +NR (MS2 ) (MS2 ) QR1 QR2 are only functions of β. In (25), the partial

(R)

(R)

derivatives of Qb1 , Qs1 , to αI are given by

(R) ∂Q2 ∂αI

and

(MS ) ∂Qb1 1 ∂αI

with respect

where the expressions of

(26)

∂P3 ∂P2 ∂αI (γR , αb1 , αs1 ), ∂αI (2, γR , ∂P1 ∂P2 ∂αs1 (2, γR α2 ) and ∂αI (2, γD1 ,

∂P1 αs1 , α2 ), ∂α (2, γR α2 ) = b1 αb1 , α2 ) are given in Appendix A due to their extensiveness. To optimize αb1 and αs1 , we should solve the equations ∂R3L2UHM = 0 and ∂R3L2UHM = 0 but these are still difficult ∂αb1 ∂αs1 VOLUME 3, 2015

Algorithm 1 Hill Climbing Algorithm (0)

(0)

Require: Initial values (αb1 , αs1 ) and step size δ ∗ , α∗ ) Ensure: Power allocation parameters (αb1 s1 1: 2: 3: 4: 5: 6: 7:

(0)

(0)

∗ , α ∗ ) ← (α , α ) (αb1 s1 b1 s1 repeat ∗ , α∗ ) (a, b) ← (αb1 s1   a0 ← a + δ · sgn ∂R3L2UHM (a, b)  ∂αb1  0 b ← b + δ · sgn ∂R3L2UHM (a, b) ∂αs1 ∗ , α ∗ ) = arg max (αb1 (x,y)∈A R3L2UHM (x, y), s1 where A = {(a, b), (a, b0 ), (a0 , b), (a0 , b0 )} ∗ , α ∗ ) = (a, b) until (αb1 s1

by sgn(x) = 1 if x > 0, sgn(x) = −1 if x < 0 and else sgn(x) = 0. The initial power allocation parameters (0) (0) (αb1 , αs1 ) are updated iteratively in the positive direction of the gradient of R3L2UHM , improving the achievable sum rate. 2) OPTIMIZING β FOR GIVEN αb1 AND αs1

Next, we optimize β for given αb1 and αs1 . By differentiating (18) with respect to β, we get (MS1 )

∂Q ∂R3L2UHM = C3 C4 R1 ∂β ∂β

(MS1 )

(MS2 (MS1 ) ∂QR1 (R)

∂QR1 ∂β )

+ C3

+ C3 QR1 (R)

(R)

∂Qb1 = N {1 − P3 (γR , αb1 , αs1 )}N −1 ∂αI ∂P3 × (γR , αb1 , αs1 ), ∂αI (R) ∂Qs1 = N {1 − P2 (2, γR , αs1 , α2 )}N −1 ∂αI ∂P2 × (2, γR , αs1 , α2 ), ∂αI (R) ∂Q2 = N {1 − P1 (2, γR α2 )}N −1 ∂αI ∂P1 × (2, γR α2 ), ∂αI (MS1 ) ∂Qb1 = N {1 − P2 (2, γD1 , αb1 , α2 )}N −1 ∂αI ∂P2 × (2, γD1 , αb1 , α2 ), ∂αI

problems as shown in Appendix A. Instead, we will make use of the hill climbing method in Algorithm 1 to determine ∂R3L2UHM is αb1 and αs1 , where ∂R3L2UHM ∂αb1 (a, b) means that ∂αb1 evaluated at the point (αs1 = a, αb1 = b), and similarly for ∂R3L2UHM ∂αs1 (a, b). The function sgn(x) in Algorithm 1 is given

∂β (R)

,

(MS2 )

QR1

(27) (MS )

1 where C3 = N 2N +NR Qb1 Qs1 Q2 and C4 = 1 + Qb1 are only functions of αb1 and αs1 . The expression of (27) and hence the analysis will differ given the two-layer HM level used in Step 2. In the sequel, we will detail the case for [BPSK]2 -HQAM, while the analysis for the case using [QPSK]2 -HQAM will be found in Appendix B. Differentiating R3L2UHM , we get ∂R3L2UHM  ∂β p NR −1 1 = C3 C4 NR 1 − erfc γR1 (1 − β) 2s  r 1 γR1 1 × − exp(−γR1 (1 − β)) 2 π 1−β r  p NR −1  −1  γR2 1 +C3 NR 1 − erfc γR2 (1 − β) π 2 2 s  NR   p 1 1 × exp(−γR2 (1 − β)) 1 − erfc γR2 β 1−β 2  p NR 1 +C3 NR 1 − erfc γR2 (1 − β) 2 s r  p NR −1 1 1 γR2 1 1 − erfc γR2 β exp(−γR2 β). × π 2 2 β (28)

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M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

The solution to ∂R3L2UHM = 0 gives a necessary condition ∂β =0 for an optimal β. We prove that solutions for ∂R3L2UHM ∂β always exist in 0 < β < 1. From (28), we obtain ∂R3L2UHM ∂β r   1 C3 C4 NR γR1 √
NR −1 1 − erfc γR1 =− 2 π 2 r   C3 NR γR2 1 √
NR −1 ×exp(−γR1 ) + N 1 − erfc γR2 2 R π 2 s ) # " ( 1 1 1 √
− exp(−γR2 ) , × 1 − erfc γR2 lim 2 β 2 β→0+ r  NR −1 ∂R3L2UHM C3 C4 NR γR1 1 lim =− ∂β 2 π 2 β→1− s ( ) 1 × lim 1−β β→1− r   C3 NR γR2 1 √
NR −1 − N 1 − erfc γR2 2 R π 2 s " ) # ( 1 1 1 √
× 1 − erfc γR2 − exp(−γR2 ) lim 2 2 β→1− 1 − β lim

β→0+

where lim f (x) and lim f (x) represent the right-hand and x→c+

x→c−

left-hand limits of a function f (x) as x approaches c. It can be seen that lim ∂R3L2UHM = +∞ > 0 and lim ∂R3L2UHM = ∂β ∂β β→0+

β→1−

−∞ < 0, thus the intermediate-value theorem guarantees the existence of at least a solution in the considered range. Although it is difficult to prove the uniqueness of the solution in (0, 1), the value β ∗ obtained by solving ∂R3L2UHM = 0 by ∂β standard numerical methods such as bisection method will be used in the proposed scheme. 3) OPTIMIZING αb1 , αs1 AND β ITERATIVELY

We adopt an iterative procedure for optimizing αb1 , αs1 and β. (0) (0) We initially set αb1 = 0.76 and αs1 = 0.19, for which 3 the corresponding [QPSK] -HQAM coincides with a uniform square 64-QAM constellation. Then, we first optimize β by determining β ∗ numerically as in Subsection IV-C.2. Next, fixing β = β ∗ we optimize αb1 and αs1 using the procedure in Subsection IV-C.1, and so on. The results in Section VI will show that despite the non-convexity of the rate/GPF expressions, these initial values guarantee a near-optimal performance for a large range of SNRs. The steps are summarized in Algorithm 2. Although it will not be developed here due to lack of space, the power ratios that maximize the GPF metric in (19) may be derived similarly. Thus, the proposed scheme with the power ratios derived for maximizing the sum-rate in (25) will be referred as the Three-Layer Two-User HM SumRate (3L2UHM-SR) scheme, while the proposed scheme derived for maximizing the GPF in (19) will be referred as the Three-Layer Two-User HM-GPF (3L2UHM-GPF) scheme. 2930

Algorithm 2 Proposed Power Allocation Optimization Algorithm (0)

(0)

Require: Initial values αb1 = 0.76, αs1 = 0.19 and iteration number L ∗ , α ∗ and β ∗ Ensure: Power allocation parameters αb1 s1 1: 2: 3: 4: 5: 6:

(0)

(0)

Obtaining β (0) for given αb1 and αs1 by using the method in Subsection IV-C.2 for i ← 1 to L do (i) (i) Obtaining αb1 and αs1 for given β (i−1) as in Subsection IV-C.1 (i) (i) Obtaining β (i) for given αb1 and αs1 as in Subsection IV-C.2 end for ∗ ← α (L) , α ∗ ← α (L) and β ∗ ← β (L) αb1 s1 s1 b1

V. PROPOSED MULTI-USER SCHEDULING ALGORITHMS

We now focus on the DL transmissions in the cellular relay system in Fig. 2, where each cell contains three sectors with one RS each, and propose sum-rate and fairness enhancing algorithms. A. SUM-RATE MAXIMIZING SCHEDULER

We first focus on sum-rate maximization and propose a scheduler based on the Proposed 3L2UHM-SR scheme where the scheme achieving the best rate among the 3L2UHM-SR scheme and the reference SU schemes is chosen. Thus, one or two users may be allocated in each frame. It will be referred as the Proposed Two User Sum-Rate Maximizing (TU-SRM) scheduler. 1. For each user k (k = 1, . . . , K ), the scheduler selects the SU scheme achieving the highest rate among DT, MH, CD and SUSC schemes in Section III. Let RSU k be the achievable rate of the user k with its best SU scheme. 2. The scheduler selects the user k ∗ with the highest rate, i.e., k ∗ = arg maxk RSU k . 3. For every user pair (i, j) where γDi ≥ γDj , the scheduler applies the Proposed 3L2UHM-SR scheme to this user pair where i, j correspond to users MS1 , MS2 in the scheme, respectively. The SIC decoding order at the RS is adapted given the SNR ordering of γDi , γRi and γRj , as explained in Section IV. Let RTU i, j given by (18) be the sum rate of user pair (i, j) served by the 3L2UHM-SR scheme. 4. The scheduler obtains the user pair (i∗ , j∗ ) with the best sum rate RTU i∗ , j∗ , (i∗ , j∗ ) = arg max RTU i,j . (i,j)

(29)

TU 5. If RSU k ∗ ≤ Ri∗ , j∗ , the scheduler allocates the current frame to the user pair (i∗ , j∗ ) simultaneously using the Proposed 3L2UHM-SR scheme. Otherwise, the user k ∗ only is served by its corresponding best-rate SU scheme. VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

B. PROPORTIONAL FAIR SCHEDULERS

To improve fairness, we propose the scheduler based on the 3L2UHM-GPF scheme where the algorithm chooses the scheme giving the best GPF metric among the 3L2UHM-GPF scheme and the SU schemes in Section III. It is referred as the Proposed Two User GPF Maximizing (TU-GPFM) scheduler. 1. For each user k (k = 1, . . . , K ), the scheduler selects the SU scheme achieving the highest rate among DT, MH, CD and SUSC schemes in Section III. 2. The scheduler selects the user k ∗ with the best NPF in (8), i.e., k ∗ = arg maxk ρkSU , and let 0kSU ∗ = log Rk ∗ . 3. For every user pair (i, j) where γDi ≥ γDj , the scheduler applies the 3L2UHM-GPF where users i, j correspond to users MS1 and MS2 in the scheme, respectively. Let 0i,TUj be the GPF of the user pair (i, j) served by 3L2UHM-GPF. 4. The scheduler determines the pair (i∗ , j∗ ) with the best GPF 0iTU ∗ , j∗ , TU . (i∗ , j∗ ) = arg max 0i,j (i,j)

(30)

5. If 0kSU ≤ 0iTU ∗ ∗ , j∗ , the scheduler allocates the current frame to the pair (i∗ , j∗ ) using the 3L2UHM-GPF. Otherwise, k ∗ only is served by its corresponding SU scheme. One drawback of the Proposed TU-GPFM scheduler is its high computational complexity due to exhaustive search over all user pairs, so we propose the low-complexity Two User Normalized PF Maximizing (TU-NPFM) scheduler: 1. For each user k (k = 1, . . . , K ), the scheduler selects the SU scheme achieving the highest rate among DT, MH, CD and SUSC schemes. Let ρkSU be user k’s NPF metric and 0kSU its GPF metric. 2. The scheduler selects the user pair (i∗ , j∗ ) with the SU best and the second best NPF metric ρiSU ∗ and ρj∗ , and applies the 3L2UHM-GPF scheme. Their GPF metric is denoted 0iTU ∗ , j∗ . 3. If 0kSU ≤ 0iTU ∗ ∗ , j∗ , the scheduler allocates the current frame to the pair (i∗ , j∗ ) using the 3L2UHM-GPF. Otherwise, k ∗ only is served by its SU scheme. VI. NUMERICAL RESULTS

First, the proposed 3L2UHM-SR and 3L2UHM-GPF schemes will be evaluated assuming the TU-RBC setting. Next, the proposed schedulers in Section V will be compared to the reference SU schedulers in Section III-B in the multiuser cellular relay system. A. TWO-USER RBC ENVIRONMENT

Fig. 7 illustrates the sum rate performance of the schemes for the given parameters and SNR values. In the TU-RBC, the reference SU-SRM scheme simply selects the single bestrate user for each set of SNRs. Here, it serves MS2 by MH VOLUME 3, 2015

for γD1 ≤ 11 dB and MS1 by DT for γD1 > 11 dB. For the proposed scheme, we compare five possible power allocation methods: • 3L2UHM-SR (cst): We fix (αb1 , αs1 ) = (0.76, 0.19), for which the constellation of [QPSK]3 -HQAM corresponds to a square 64-QAM constellation. Similarly, for having a square constellation, we set β = 0.2 for [QPSK]2 -HQAM in Step 2. • 3L2UHM-SR (opt.β): (αb1 , αs1 ) = (0.76, 0.19) and optimal β obtained by exhaustive search. • 3L2UHM-SR (opt.α): with optimal αb1 , αs1 obtained by exhaustive search and β = 0.2. • 3L2UHM-SR (exh): with optimal power allocation parameters obtained by exhaustive search. ∗ , α ∗ and β ∗ obtained • 3L2UHM-SR (num): with αb1 s1 numerically based on the iterative algorithm in Subsection IV-C.3, setting L = 1.

FIGURE 7. Sum rate performance of the proposed and reference schemes with N = 12, γR = 40 dB, γR1 = 10 dB, γR2 = 20 dB.

Fig. 7 shows that the proposed 3L2UHM-SR (num) closely approaches 3L2UHM-SR (exh), the optimal sum rate given by exhaustive search, showing the validity of our scheme. For these SNRs, optimizing β in 3L2UHM-SR (opt.β) has a greater impact on the sum rate compared to optimizing αb1 and αs1 in 3L2UHM-SR (opt.α). This is due to the low value of γR1 , so that β should be optimized to reduce the decoding errors for xs1 . The large gains of 3L2UHM-SR (num) over 3L2UHM-SR (cst) illustrate the necessity of the proposed power optimization. Moreover, the proposed scheme always outperforms the reference 2L2USC, as 3L2UHM-SR (num) enables to make efficient use of the high quality relayed link to MS1 , thanks to the three-layered HM signal from BS. Compared to SU-SRM, our scheme achieves a higher rate in the region 3.7 ≤ γD1 ≤ 12.7 dB. Thus, the best strategy here is to use MH to MS2 for 0 < γD1 < 3.7 dB, the proposed 3L2UHM-SR (num) for 3.7 ≤ γD1 ≤ 12.7 dB, then DIR to MS1 for 12.7 dB < γD1 with high modulation levels (16-QAM and 64-QAM) given the high direct link quality. 2931

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

FIGURE 8. Sum rate of the proposed and reference schemes with N = 12, γR = 40 dB, γR1 = 20 dB, γR2 = 15 dB.

FIGURE 10. GPF metric for proposed and reference schemes with PF with N = 12, γR = 40 dB, γR1 = 20 dB, γR2 = 15 dB.

Fig. 8 shows the sum rate performance for a different set of SNRs. Again, 3L2UHM-SR (num) closely matches the optimal performance 3L2UHM-SR (exh). In this case, since both γR1 and γR2 have higher values, we get a larger impact by optimizing αb1 and αs1 than β, as shown by 3L2UHM-SR (opt.α) and 3L2UHM-SR (opt.β). Here, the reference SU-SRM serves MS1 by CD for γD1 < 12.9 dB, then by DT for 12.9 dB < γD1 . Moreover, 3L2UHM-SR (num) outperforms SU-SRM over 3.5 ≤ γD1 ≤ 12.9 dB, so the best strategy is CD to MS1 for 0 < γD1 < 3.5 dB, 3L2UHM-SR (num) for 3.5 ≤ γD1 ≤ 12.9 dB, and then DT to MS1 for γD1 > 12.9 dB.

the sum rate and GPF very close to the optimal ones given by 3L2UHM-GPF (exh). Compared to the proposed scheme designed for sum-rate maximization 3L2UHM-SR (num), 3L2UHM-GPF (num) achieves a slightly lower sum-rate but larger GPF metric, showing the effectiveness of the proposed GPF maximizing power allocation. Moreover, 3L2UHM-GPF (num) significantly outperforms that with constant power ratios 3L2UHM-GPF (cst), both in terms of sum-rate and GPF metric. For GPF, 3L2UHM-GPF (num) outperforms SU-NPFM over 2.7 ≤ γD1 ≤ 13.9 dB. Thus, for GPF maximization, one should use the CD scheme to MS1 for 0 < γD1 < 2.7 dB, 3L2UHM-GPF (num) for 3.5 ≤ γD1 ≤ 13.9 dB, then the DT scheme to MS1 for 13.9 dB < γD1 . These simulations have confirmed the validity of the proposed method in Algorithm 2, as it achieves near-optimal performance over a large range of SNRs despite the complexity and non-convexity of the sum-rate/GPF expressions. In addition, the proposed method considerably reduces the required computational complexity compared to exhaustive search, as a near-optimal performance with only L = 1 iteration can be achieved using standard numerical methods. Moreover, as we assume only static or low mobility scenarios, the power optimization only needs to be performed when the channels vary significantly, i.e., typically every few frames. B. MULTI-USER SCHEDULING

FIGURE 9. Sum rate of proposed and reference schemes with PF with N = 12, γR = 40 dB, γR1 = 20 dB, γR2 = 15 dB.

Next, the Proposed 3L2UHM-GPF scheme is evaluated for N = 12, γR = 40 dB, γR1 = 20 dB, γR2 = 15 dB in Figs. 9 and 10 in terms of sum-rate and GPF metric, respectively. The benchmark SU-NPFM scheme selects the user with the best NPF metric among the two users, for each set of SNRs. Proposed 3L2UHM-GPF (num) achieves 2932

In this section, we evaluate the different algorithms in a multiuser cellular environment. The simulation parameters are set as follows. The radius of the cell is fixed to D = 1000 m, and the BS-RS distance to DR = 600 m. The fixed RS is deployed so that a high quality BS-RS link in Line-Of-Sight is guaranteed as recommended in [3], hence it is modeled as an AWGN channel with an average SNR of 40 dBs. All the other channels undergo Rayleigh fading. The average SNR γ¯k for MSk is given by γ¯k = P( dDk )µ where dk is the distance of MSk to the BS or RS, and µ = 3 the path loss exponent, VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

which is a typical value for outdoor environments [38]. BS transmit power P is given such that a received SNR level of 5 dBs at the cell edge is ensured, and the RS transmit power PR is equal to P.

both metrics. Note that, even if only sum-rate maximization is aimed by TU-SRM scheduler, it also greatly improves the fairness level as well, as shown in Fig. 12. This is because in the Proposed 3L2UHM-SR scheme, any user having either a good direct or relayed link gets a higher chance to be selected as one user of the allocated pair, while only users experiencing high quality over both direct and relayed links are favored by reference SU schedulers. Moreover, despite its suboptimality, the proposed TU-NPFM scheduler still outperforms the reference SU-NPFM scheduler both in terms of sum-rate and GPF metric. This performance gain is achieved with much lower computational complexity compared to the Proposed TU-GPFM scheduler, which is quite promising for practical implementation. VII. CONCLUSION

FIGURE 11. Sum rate performance of proposed and reference schedulers.

FIGURE 12. GPF performance of proposed and reference schedulers.

In Figs. 11 and 12, we compare the sum rate and GPF metric of the proposed and reference schedulers against the number of users K per sector, respectively. First of all, we observe that all the proposed algorithms outperform the reference SU-NPFM scheduler of Section III-B that allocates a single user per scheduling frame. This is because the proposed schedulers enable to make use of the best combination of high quality direct and relayed links that may belong to two different users. Moreover, the Proposed 3L2UHM-SR and 3L2UHM-GPF schemes underlying those schedulers can take full advantage of the available links thanks to the optimized power ratios among the three HM layers. As expected, the proposed TU-SRM scheduler achieves a higher sum-rate than the proposed TU-GPFM scheduler, and vice versa concerning the GPF performance. Nevertheless, both schedulers largely outperform the reference SU-NPFM scheduler for VOLUME 3, 2015

We have proposed SC-based relaying schemes and non-orthogonal schedulers designed under the practical constraints of discrete HMs. In the proposed schemes, the messages destined to two users are superposed into three HM layers with given power ratios that are tuned to the qualities of the corresponding three links, i.e., the best direct and the two relayed links. We have first derived the sumrate and GPF analytical expressions, taking into account the decoding errors for all messages at each step. Given the intractability of the joint power optimization problem, we have instead proposed a sub-optimal power allocation based on iterative optimization and numerical methods. This gave rise to the 3L2UHM-SR scheme for sum-rate maximization and the 3L2UHM-GPF scheme for GPF maximization. Next, these schemes were integrated into two schedulers for the DL of a multi-user cellular relay system, the TU-SRM and TU-GPFM schedulers for sum-rate and GPF enhancement, respectively. A low-complexity scheduler based on PF, TU-NPFM, was also proposed. The simulation results have shown that both 3L2UHM-SR and 3L2UHM-GPF schemes with the proposed power allocation closely approached the optimal performance. Moreover, these schemes improved the rate and GPF metrics compared to conventional relaying schemes over a large SNR region. Finally, in the multi-user setting, the proposed schedulers outperformed conventional orthogonal schedulers both in terms of sum-rate and fairness. More importantly, we could confirm that the main conclusions of previous works such as [15] and [17] assuming Gaussian channels were also true under the more stringent constraints of discrete HMs, namely the superiority of nonorthogonal schedulers over conventional orthogonal schedulers for various system level metrics. These conclusions open up new perspectives for the design and integration of non-orthogonal multiple access into next-generation cellular relay systems. In the future work, channel coding will be considered in conjunction of the discrete HM levels in the proposed schemes, and the impact of adaptive discrete HM and coding on the different schemes and schedulers will be evaluated. 2933

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APPENDIX A

Below are given all the partial derivatives’ expressions involved in (25)-(26) for sum-rate optimization in IV-C. ∂P3 (γR , αb1 , αs1 ) ∂αb1 r 1 γR = 4 2π s ! ( s 1 1 × + αb1 α2 r r r 2 ! γR αb1 γR αs1 γR α2 × exp − − − 2 2 2 s ! s 1 1 + − αb1 α2 r r r 2 ! γR αb1 γR αs1 γR α2 × exp − − + 2 2 2 s ! s 1 1 + + αb1 α2 r r 2 ! r γR αb1 γR αs1 γR α2 + − × exp − 2 2 2 s ! s 1 1 + − αb1 α2 r r r 2 !) γR αb1 γR αs1 γ R α2 × exp − + + 2 2 2 r r  r  1 γR αb1 γR αs1 γ R α2 × 1 − erfc − − 8 2 2 2 r r r  γR αb1 γR αs1 γR α2 1 − + − erfc 8 2 2 2 r r r  1 γR αb1 γR αs1 γR α2 + − − erfc 8 2 2 2 r r r  1 γR αb1 γR αs1 γR α2 − erfc + + , 8 2 2 2 ∂P3 (γR , αb1 , αs1 ) ∂αs1 r 1 γR = 4 2π s ! ( s 1 1 × + αs1 α2 r r r 2 ! γR αb1 γR αs1 γR α2 × exp − − − 2 2 2 s ! s 1 1 + − − αs1 α2 r r r 2 ! γR αb1 γR αs1 γR α2 × exp − − + 2 2 2 2934

s ! 1 1 + + αs1 α2 r r r 2 ! γR αb1 γR αs1 γR α2 × exp − + − 2 2 2 s ! s 1 1 + − αs1 α2 r r r 2 !) γR αb1 γR αs1 γ R α2 × exp − + + 2 2 2 r r   r 1 γR αb1 γR αs1 γ R α2 × 1 − erfc − − 8 2 2 2 r r  r 1 γR αb1 γR αs1 γ R α2 − + − erfc 8 2 2 2 r r r  1 γR αb1 γR αs1 γ R α2 − erfc + − 8 2 2 2 r r r  γR αb1 γR αs1 γ R α2 1 + + , − erfc 8 2 2 2 s

∂P1 (2, γR α2 ) ∂αb1 ∂P1 = (2, γR α2 ) ∂αs1 r  r  γ α  γR γ R α2 1 R 2 =− exp − 1 − erfc , 2π α2 2 2 2 ∂P2 (2, γR , αs1 , α2 ) ∂αb1 (s r r r 2 ! 1 γR αs1 γR α2 1 γR = × exp − − 2 π 2α2 2 2 s r r 2 !) 1 γR αs1 γR α2 + − exp − 2α2 2 2 r  r  1 γR αs1 γR α2 × 1 − erfc − 4 2 2 r r  1 γR αs1 γR α2 × − erfc + , 4 2 2 ∂P2 (2, γR , αs1 , α2 ) ∂αs1 r 1 γR = 2 π s ( s ! r r 2 ! 1 1 γR αs1 γ R α2 × + exp − − 2αs1 2α2 2 2 s s ! r r 2 !) 1 1 γR αs1 γ R α2 + − exp − + 2αs1 2α2 2 2 r  r  1 γR αs1 γR α2 × 1 − erfc − 4 2 2 r r  1 γR αs1 γR α2 × − erfc + , 4 2 2 VOLUME 3, 2015

M. Kaneko et al.: Fairness-Aware Non-Orthogonal Multi-User Access With Discrete HM

∂P2 (2, γD1 , αb1 , α2 ) ∂αb1 r r   r 1 1 γD1 γD1 αb1 γD1 α2 1 − erfc = − 2 2π 4 2 2 r r   1 γD1 αb1 γD1 α2 − erfc + 4 2 2 s ! ( s r r 2 ! 1 1 γD1 αb1 γD1 α2 × + exp − − αb1 α2 2 2 s ! s r 2 !) r 1 1 γD1 αb1 γD1 α2 + , exp − − + αb1 α2 2 2

!)2NR r 1 γR2 β 1 − erfc 2 2 ! r r 1 γR2 (1 − β) γR2 β × 1 − erfc − 4 2 2 !)2NR −1 r r 1 γR2 (1−β) γR2 β + − erfc 4 2 2 s ! ( s 1 1 × + 2(1 − β) 2β  !2  r r γ (1 − β) γ β R2 R2  × exp− − 2 2 s ! s 1 1 − + 2(1 − β) 2β  !2  r r γR2 (1−β) γR2 β  + ×exp−  2 2 ( !)2NR −1 r r γR2 γR2 β 1 + C1 NR 1 − erfc π 2 2 s   1 γR2 β exp − × 2β 2 ( ! r r γR2 (1 − β) γR2 β 1 × 1 − erfc − 4 2 2 !)2NR r r γR2 (1 − β) γR2 β 1 . − erfc + 4 2 2

C1 NR F2 (β) = − 2 (

∂P2 (2, γD1 , αb1 , α2 ) ∂αs1 r r   r 1 γD1 αb1 γD1 α2 1 γD1 1 − erfc − = 2 2π 4 2 2 r r  1 γD1 αb1 γD1 α2 − erfc + 4 2 2 (s r r 2 ! 1 γD1 αb1 γD1 α2 × − exp − α2 2 2 s r r  2 !) 1 γD1 αb1 γD1 α2 − exp − + . α2 2 2 APPENDIX B

We provide the analysis for optimizing β given αb1 , αs1 in (18) as in Section IV-C.2 for HM level [QPSK]2 -HQAM and [8-PAM]2 -HQAM. A. [QPSK ]2 -HQAM IN STEP 2

With [QPSK]2 -HQAM in Step 2, we have 0 < β < 1/2, ∂R3L2UHM = F1 (β) + F2 (β), where ∂β r C1 C2 NR γR1 F1 (β) = − 2 π ! ( r r 1 γR1 (1 − β) γR1 β − × 1 − erfc 4 2 2 !)2NR −1 r r 1 γR1 (1−β) γR1 β − erfc + 4 2 2 s ! ( s 1 1 × + 2(1 − β) 2β  !2  r r γ (1 − β) γ β R1 R1  − ×exp− 2 2 s s ! 1 1 + − 2(1 − β) 2β  r !2  r γR1 (1−β) γR1 β  × exp− , +  2 2 VOLUME 3, 2015

r

γR2 π

(

From which we obtain ∂R3L2UHM ∂β r  γ  C1 C2 NR γR1 R1 =− √ exp − π 2 2 2  r 2NR −1 1 γR1 × 1 − erfc 2 2 r  r 2NR −1  2NR −1 γR2 1 γR2 1 1 − erfc +C1 NR π 2 2 2 s ) " r  ( 1 γR2 1 × 1 − erfc lim + 2 2 2β β→0 #  γ  1 R2 − √ exp − , 2 4 2

lim

β→0+

∂R3L2UHM ∂β r γR1 = −C1 C2 NR π   3 1 √
2NR −1 × − erfc γR1 4 4

lim

− β→ 21

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 3 1 √
2NR −1 −C1 NR − erfc γR2 4 4 2NR −1  r 1 γR2 × 1 − erfc 2 4     r 1 γR2 3 1 √
× 1 − erfc − − erfc γR2 2 4 4 4  γ  R2 . × exp − 4 r

γR2 π



For analyzing these limits, we define the following function of x, r     x √
3 1 1 x − f (x) = 1 − erfc − erfc x exp − . 2 4 4 4 4 Differentiating f (x) with respect to x, we have  x df (x) 1 = exp − dx 4 4 "r # √
1 3 1 × (1 − exp(−x)) + − erfc x . πx 4 4 Since dfdx(x) > 0 for x > 0 and f (0) = 0, f (γR2 ) is always < 0. Moreover, we see that positive. Thus, lim ∂R3L2UHM ∂β lim

β→0+

∂R3L2UHM ∂β

β→ 21

−

= +∞ > 0. Therefore, the intermediate-

value theorem guarantees the existence of at least a solution in the considered range. Again, the proposed scheme chooses the value β ∗ obtained by solving ∂R3L2UHM = 0 numerically. ∂β

We obtain ∂R3L2UHM ∂β r  γ  7C1 C2 NR γR1 R1 exp − =− 8 21π 21  NR −1 r 7 γR1 × 1 − erfc 8 21 r  r NR −1  NR −1 7C1 NR γR2 7 γR2 1 + 1 − erfc 8 21π 8 21 8 s ) # " ( r  γ  γR2 1 1 7 R2 lim − exp − , × 1− erfc 8 21 8 21 β→0+ β lim

β→0+

∂R3L2UHM ∂β s ( ) r   7C1 C2 NR γR1 1 NR −1 1 =− lim 8 21π 8 1−β β→1− r r  NR −1  NR −1 7C1 NR γR2 γR2 1 7 − 1 − erfc 8 21π 8 21 8 s ) " # r (  γR2 1 1 γR2  7 exp − × 1− erfc lim . 8 21 8 21 β→1− 1−β lim

β→1−

We have lim

β→0+

∂R3L2UHM ∂β

= +∞ > 0, lim

β→1−

∂R3L2UHM ∂β

=

B. [8-PAM] 2 -HQAM IN STEP 2

−∞ < 0, so the intermediate-value theorem guarantees the existence of a solution in the considered range. Similarly to the other cases, we use the value β ∗ obtained by solving ∂R3L2UHM = 0 numerically. ∂β

In this case, we have 0 < β < 1 and

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∂R3L2UHM ∂β !)NR −1 r 7 γR1 (1 − β) = C1 C2 NR 1 − erfc 8 21 s  r   7 γR1 1 γR1 (1 − β) × − exp − 8 π 21(1 − β) 21 ( !)NR −1 r r γR2 7 γR2 (1 − β) +C1 NR 1 − erfc π 8 21 s     −7 1 γR2 (1 − β) × exp − 8 21(1 − β) 21 ( !)NR r 7 γR2 β × 1 − erfc 8 21 r  √ NR γR2 7 γR2 (1 − β) +C1 NR 1 − erfc π 8 21 ( !)NR −1 s r   7 γR2 β 7 1 γR2 β × 1 − erfc exp − . 8 21 8 21β 21 (

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MEGUMI KANEKO (M’05) received the B.S. and M.Sc. degrees in communication engineering from the Institut National des Télécommunications, France, in 2003 and 2004, respectively, the M.Sc. degree from Aalborg University, Denmark, and the Ph.D. degree from Aalborg University, in 2007. In 2007, she was a Visiting Researcher with Kyoto University, Kyoto, Japan, and a JSPS Post-Doctoral Fellow from 2008 to 2010. She is currently an Assistant Professor with the Department of Systems Science, Graduate School of Informatics, Kyoto University. Her research interests include wireless communication, cross-layer protocol design, and communication theory. She received the 2009 Ericsson Young Scientist Award, the IEEE Globecom 2009 Best Paper Award, the 2011 Funai Young Researcher’s Award, the WPMC 2011 Best Paper Award, and the 2012 Telecom System Technology Award.

HIROFUMI YAMAURA received the B.E. degree in applied mathematics and physics and the M.E. degree in systems science from the Department of Systems Science, Graduate School of Informatics, Kyoto University, Japan, in 2011 and 2013, respectively. His research interests include signal processing and radio resource allocation in wireless communication systems. He received the IEEE VTC2011-Fall Young Researcher’s Encouragement Award from the IEEE VTS Japan Chapter.

YOUHEI KAJITA received the B.E. degree from Osaka Prefecture University, in 2012, and the M.E. degree in systems science from the Department of Systems Science, Graduate School of Informatics, Kyoto University, Japan, in 2014. His research interests include signal processing and radio resource allocation in wireless communication systems.

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KAZUNORI HAYASHI (M’97) received the B.E., M.E., and Ph.D. degrees in communication engineering from Osaka University, Osaka, Japan, in 1997, 1999, and 2002, respectively. Since 2002, he has been with the Department of Systems Science, Graduate School of Informatics, Kyoto University, where he is currently an Associate Professor. His research interests include statistical signal processing for communication systems. He is a member of IEICE and ISCIE. He received the ICF Research Award from the KDDI Foundation in 2008, the IEEE Globecom 2009 Best Paper Award, the IEICE Communications Society Best Paper Award in 2010, the WPMC’11 Best Paper Award, the Telecommunications Advancement Foundation Award in 2012, and the IEICE Communications Society Best Tutorial Paper Award in 2013.

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HIDEAKI SAKAI (LF’14) received the B.E. and D.E. degrees in applied mathematics and physics from Kyoto University, Kyoto, Japan, in 1972 and 1981, respectively. He spent six months with Stanford University as a Visiting Scholar from 1987 to 1988. He has been a Professor with the Department of Systems Science, Graduate School of Informatics, Kyoto University, where he is currently a Professor Emeritus. His research interests are in the areas of adaptive and statistical signal processing. He served twice as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING.

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